1 Magnetohydrodynamics ......................................... 1
1.1 The Fluid Approximation ................................. 2
1.1.1 Matter as a Continuum ............................ 2
1.1.2 Solid Versus Fluid ............................... 3
1.2 Essentials of Hydrodynamics ............................. 4
1.2.1 Mass: The Continuity Equation .................... 4
1.2.2 The D/Dt Operator ................................ 6
1.2.3 Linear Momentum: The Navier-Stokes Equations ..... 7
1.2.4 Angular Momentum: The Vorticity Equation ........ 10
1.2.5 Energy: The Entropy Equation .................... 12
1.3 The Magnetohydrodynamical Induction Equation ........... 13
1.4 Scaling Analysis ....................................... 15
1.5 The Lorentz Force ...................................... 18
1.6 Joule Heating .......................................... 20
1.7 The Full Set of MHD Equations .......................... 20
1.8 MHD Waves .............................................. 22
1.9 Magnetic Energy ........................................ 23
1.10 Magnetic Flux Freezing and Alfven's Theorem ............ 24
1.11 The Magnetic Vector Potential .......................... 25
1.12 Magnetic Helicity ...................................... 26
1.13 Force-Free Magnetic Fields ............................. 26
1.14 The Ultimate Origin of Astrophysical Magnetic Fields ... 27
1.14.1 Why В and not E? ............................... 27
1.14.2 Monopoles and Batteries ........................ 28
1.15 The Astrophysical Dynamo Problem(s) .................... 30
1.15.1 A Simple Dynamo ..................................... 30
1.15.2 The Challenges ...................................... 33
Bibliography ................................................ 34
2 Decay and Amplification of Magnetic Fields .................. 37
2.1 Resistive Decays of Magnetic Fields .................... 37
2.1.1 Axisymmetric Magnetic Fields .................... 38
2.1.2 Poloidal Field Decay ............................ 39
2.1.3 Toroidal Field Decay ............................ 41
2.1.4 Results for a Magnetic Diffusivity Varying
with Depth ...................................... 42
2.1.5 Fossil Stellar Magnetic Fields .................. 43
2.2 Magnetic Field Amplification by Stretching and
Shearing ............................................... 44
2.2.1 Hydrodynamical Stretching and Field
Amplification ................................... 44
2.2.2 The Vainshtein & Zeldovich Flux Rope Dynamo ..... 46
2.2.3 Hydrodynamical Shearing and Field
Amplification ................................... 48
2.2.4 Toroidal Field Production by Differential
Rotation ........................................ 48
2.3 Magnetic Field Evolution in a Cellular Flow ............ 52
2.3.1 A Cellular Flow Solution ........................ 52
2.3.2 Flux Expulsion .................................. 57
2.3.3 Digression: The Electromagnetic Skin Depth ...... 59
2.3.4 Timescales for Field Amplification and Decay .... 60
2.3.5 Flux Expulsion in Spherical Geometry:
Axisymmetrization ............................... 62
2.4 Two Anti-Dynamo Theorems ............................... 64
2.4.1 Zeldovich's Theorem ............................. 65
2.4.2 Cowling's Theorem ............................... 66
2.5 The Roberts Cell Dynamo ................................ 68
2.5.1 The Roberts Cell ................................ 68
2.5.2 Dynamo Solutions ................................ 69
2.5.3 Exponential Stretching and Stagnation Points .... 72
2.5.4 Mechanism of Field Amplification in the
Roberts Cell .................................... 73
2.5.5 Fast Versus Slow Dynamos ........................ 74
2.6 The CP Flow and Fast Dynamo Action ..................... 75
2.6.1 Dynamo Solutions ................................ 76
2.6.2 Fast Dynamo Action and Chaotic Trajectories ..... 78
2.6.3 Magnetic Flux Versus Magnetic Energy ............ 80
2.6.4 Fast Dynamo Action in the Nonlinear Regime ...... 81
2.7 Dynamo Action in Turbulent Flows ....................... 82
Bibliography ................................................ 83
3 Dynamo Models of the Solar Cycle ............................ 87
3.1 The Solar Magnetic Field ............................... 88
3.1.1 Sunspots and the Photospheric Magnetic Field .... 88
3.1.2 Hale's Polarity Laws ............................ 90
3.1.3 The Magnetic Cycle .............................. 92
3.1.4 Sunspots as Tracers of the Sun's Internal
Magnetic Field .................................. 93
3.1.5 A Solar Dynamo Shopping List .................... 94
3.2 Mean-Field Dynamo Models ............................... 95
3.2.1 Mean-Field Electrodynamics ...................... 95
3.2.2 The α-Effect .................................... 97
3.2.3 Turbulent Pumping .............................. 101
3.2.4 The Turbulent Diffusivity ...................... 102
3.2.5 The Mean-Field Dynamo Equations ................ 103
3.2.6 Dynamo Waves ................................... 103
3.2.7 The Axisymmetric Mean-Field Dynamo Equations ... 105
3.2.8 Linear αΩ Dynamo Solutions ..................... 107
3.2.9 Nonlinearities and α-Quenching ................. 112
3.2.10 Kinematic αΩ Models with α-Quenching ........... 113
3.2.11 Enters Meridional Circulation: Flux
Transport Dynamos .............................. 116
3.2.12 Interface Dynamos .............................. 118
3.3 Babcock-Leighton Models ............................... 121
3.3.1 Sunspot Decay and the Babcock-Leighton
Mechanism ...................................... 122
3.3.2 Axisymmetrization Revisited .................... 127
3.3.3 Dynamo Models Based on the Babcock-Leighton
Mechanism ...................................... 128
3.3.4 The Babcock-Leighton Poloidal Source Term ...... 129
3.3.5 A Sample Solution .............................. 130
3.4 Models Based on HD and MHD Instabilities .............. 132
3.4.1 Models Based on Shear Instabilities ............ 132
3.4.2 Models Based on Flux-Tube Instabilities ........ 134
3.5 Global MHD Simulations ................................ 135
3.6 Local MHD Simulations ................................. 143
Bibliography ............................................... 146
4 Fluctuations, Intermittency and Predictivity .............. 153
4.1 Observed Patterns of Solar Cycle Variations ........... 153
4.1.1 Pre-Telescopic and Early Telescopic Sunspot
Observations ................................... 153
4.1.2 The Sunspot Cycle .............................. 155
4.1.3 The Butterfly Diagram .......................... 156
4.1.4 The Waldmeier and Gnevyshev-Ohl Rules .......... 158
4.1.5 The Magnetic Activity Cycle .................... 160
4.1.6 The Maunder Minimum ............................ 160
4.1.7 From Large-Scale Magnetic Fields to Sunspot
Number ......................................... 162
4.2 Cycle Modulation Through Stochastic Forcing ........... 164
4.3 Cycle Modulation Through the Lorentz Force ............ 168
4.4 Cycle Modulation Through Time Delays .................. 171
4.5 Intermittency ......................................... 173
4.6 Model-based Cycle Predictions ......................... 176
4.6.1 The Solar Polar Magnetic Field as a Precursor .. 177
4.6.2 Model-Based Prediction Using Solar Data ........ 180
Bibliography ............................................... 182
5 Stellar Dynamos ............................................ 187
5.1 Early-Type Stars ...................................... 189
5.1.1 Mean-Field Models .............................. 189
5.1.2 Numerical Simulations of Core Dynamo Action .... 192
5.1.3 Getting the Magnetic Field to the Surface ...... 194
5.1.4 Alternative to Core Dynamo Action .............. 194
5.2 A-Type Stars .......................................... 195
5.2.1 Observational Overview ......................... 195
5.2.2 The Fossil Field Hypothesis .................... 197
5.2.3 Dynamical Stability of Large-Scale Magnetic
Fields ......................................... 197
5.2.4 The Transition to Solar-Like Dynamo Activity ... 197
5.3 Solar-Type Stars ...................................... 198
5.3.1 Observational Overview ......................... 198
5.3.2 Empirical Stellar Activity Relationships ....... 199
5.3.3 Solar and Stellar Spin-Down .................... 200
5.3.4 Modelling Dynamo Action in Solar-Type Stars .... 206
5.4 Fully Convective Stars ................................ 207
5.5 Pre- and Post-Main-Sequence Stars ..................... 208
5.6 Compact Objects ....................................... 209
5.7 Galaxies and Beyond ................................... 210
Bibliography ............................................... 211
Appendix A: Useful Identities and Theorems from Vector
Calculus ................................................... 215
Appendix B: Coordinate Systems and the Fluid Equations ........ 217
Appendix C: Physical and Astronomical Constants ............... 227
Appendix D: Maxwell's Equations and Physical Units ............ 229
Index ......................................................... 233
|
